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Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

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In this paper, we employ the periodic unfolding method for simulating the electromagnetic field in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the effective parameters for a Debye dielectric medium in the case of a circular microstructure in two dimensions. We assume that the composite materials are quasi-static in nature, i.e., the wavelength of the electromagnetic field is much larger than the relevant dimensions of the microstructure.

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References

  1. Allaire G. (1992). Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6):1482–1518

    Article  MATH  MathSciNet  Google Scholar 

  2. Banks, H. T., Bokil, V. A., Cioranescu, D., Gibson, N. L., Griso, G., and Miara, B. (2005). Homogenization of periodically varying coefficients in electromagnetic materials. CRSC Technical Report, CRSC-TR05-05.

  3. Banks, H. T., Buksas, M. W., and Lin, T. (2000). Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts, Frontiers in Applied Mathematics. v. FR21, SIAM, Philadelphia, PA, 2000.

  4. Banks, H. T., and Gibson, N. L. (2005). Inverse problems for Maxwell’s equations with distributions of dielectric parameters. CRSC Technical Report CRSC-TR05-29, August 2005, N.C. State University; available at http://www.ncsu.edu/crsc/

  5. Banks H.T., Gibson N.L., and Winfree W.P. (2005). Gap detection with electromagnetic Terahertz signals. Nonlin. Anal. Real World Appl. 6:381–416

    Article  MATH  MathSciNet  Google Scholar 

  6. Bossavit A., Griso G., and Miara B. (2004). Modélisation de structures électromagnetiques périodiques: matériaux bianisotropiques avec mémoire. C. R. Acad. Sci. Paris, Ser. I. 338:97–102

    MATH  MathSciNet  Google Scholar 

  7. Bossavit, A., Griso, G., and Miara, B., Modelling of periodic electromagnetic structures bianisotropic material with memory effects, J. Mathématiques Pures et Appliqu’ees, in press.

  8. Cioranescu, D. (2003). Homogenization and applications to material sciences in Research Directions in Distributed Parameter Systems, Frontiers in Applied Mathematics, v. FR27, SIAM, Philadelphia, PA.

  9. Cioranescu D., Damlamian A., and Griso G. (2002). Periodic unfolding and homogenization. C. R. Acad. Sci. Paris, Ser. I 335:99–104

    MATH  MathSciNet  Google Scholar 

  10. Cioranescu, D., and Donato, P. (1999). An Introduction to Homogenization, Number 17 in Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford.

  11. Cole K.S., and Cole R.H. (1941). Dispersion and absorption in dielectrics. J. Chem. Phy. 9:341–351

    Article  CAS  Google Scholar 

  12. Debye P. (1929). Polar Molecules Chemical Catalog Co., New York

    MATH  Google Scholar 

  13. Engström, C., and Sjöberg, D., (2004). A comparison of two numerical methods for homogenization of Maxwell’s equations, Technical Report LUTEDX/(TEAT-7121)/1-10/ (2004), Department of Electroscience, Lund Institute of Technology, Sweden.

  14. El Feddi M., Ren Z., Razek A., and Bossavit A. (1997). Homogenization technique for Maxwell equations in periodic structures. IEEE Trans. Mag. 33(2):1382–1385

    Article  Google Scholar 

  15. Gibson N.L. (2004). Terahertz-Based Electromagnetic Interrogation Techniques for Detection. Ph.D. Thesis, N.C. State University, Raleigh, NC

    Google Scholar 

  16. Kelley D.F., and Luebbers R.J. (1996). Piecewise linear recursive convolution for dispersive media using FDTD. IEEE Trans. Antennas Propagat. 44(6):792–797

    Article  Google Scholar 

  17. Kristensson, G. (2001). Homogenization of the Maxwell equations in an anisotropic material Technical Report LUTEDX/(TEAT-7104)/1-12/(2001), Department of Electroscience, Lund Institute of Technology, Sweden, 2001.

  18. Kristensson, G. (2004). Homogenization of corrugated interfaces in electomagnetics Technical Report LUTEDX/(TEAT-7122)/1-29/(2004), Dept. of Electroscience, Lund Institute of Technology, Sweden.

  19. Luebbers R.J., Hunsberger F.P., Kunz K.S., Standler R.B., and Schneider M. (1990). A frequency-dependent finite difference time-domain formulation for dispersive materials. IEEE Trans. Elect. Compat. 32(3):222–227.

    Article  Google Scholar 

  20. NASA Facts: Thermal Protection System. NASA Report FS-2004-08-97 MSFC, NASA Marshall Space Flight Center, Huntsville, AL.

  21. Nguetseng G. (1989). A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3):608–623

    Article  MATH  MathSciNet  Google Scholar 

  22. Ouchetto, O., Zouhdi, S., Bossavit, A., Griso G., and Miara, B. (2005). Homogenization of 3D structured composites of complex shaped inclusions, PIERS 2005. In Progress in Electromagnetics Research Symposium, pp. 22–260. Hangzhou. China.

  23. Sihvola, A. (1999). Electromagnetic Mixing Formulas and Applications, IEE Electromagnetic Waves Series, The Institute of Electrical Engineers, London, p. 47.

  24. Sihvola A. (2000). Effective permittivity of mixtures: numerical validation by the FDTD method. IEEE Trans. Geosci. Remote Sensing 38(3):1303–1308

    Article  Google Scholar 

  25. Sjöberg, D. (2004). Homogenization of dispersive material parameters for Maxwell’s equations using a singular value decomposition. Technical Report LUTEDX/(TEAT-7124)/ 1-24/(2004), Department of Electroscience, Lund Institute of Technology, Sweden.

  26. Sjöberg, D., Engström, C., Kristensson, G., Wall, D. J. N., and Wellander, N. (2003).A Floquet-Bloch decomposition of Maxwell’s equations, applied to homogenization Technical Report LUTEDX/(TEAT-7119)/1-27/(2003), Department of Electroscience, Lund Institute of Technology, Sweden.

  27. Wellander, N., and Kristensson, G. (2002). Homogenization of the Maxwell equations at fixed frequency. Technical Report LUTEDX/(TEAT-7103)/1-38/(2001), Department of Electroscience, Lund Institute of Technology, Sweden.

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Banks, H.T., Bokil, V.A., Cioranescu, D. et al. Homogenization of Periodically Varying Coefficients in Electromagnetic Materials. J Sci Comput 28, 191–221 (2006). https://doi.org/10.1007/s10915-006-9091-y

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  • DOI: https://doi.org/10.1007/s10915-006-9091-y

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